Spie Press Book

This book provides a faithful and robust simulation of the optical and visual performances of the human eye for axial vision of distant objects in a variety of visual conditions. The author moves from intrinsically theoretical aspects (the optical and neurophysical models of the eye) to include a great number of experimental measurements from the scientific literature, in order to adapt the model parameters to the observed phenomenology and validate the predictivity power of the models themselves. The results are very satisfactory in terms of quantitative and qualitative adherence of model predictions to field measurements.

Resulting from the author's investigations over the last decade, the book material is largely original, and the most relevant achievement can be found in the capacity to evaluate visual acuity for a range of visual conditions, such as variations in pupil size, refractive error, and ambient illumination.

Thanks to the general organization of the book, chapters and paragraphs with high level mathematical and physical optics content can be safely skipped without compromising the overall comprehension. To this end, a brief summary is provided at the end of each chapter, making this book appropriate for readers with greatly varying degrees of technical knowledge.

• What is the typical visual performance of an average human eye on axis in different visual conditions?

• Which are the most relevant optical factors limiting human visual performance

• What is the ultimate visual performance of the eye?

• Is it possible to enhance the visual performance of the human eye through either optical aids or surgery?

• Is the existence of spatial frequency channels compatible with the neurophysical model here developed?

These basic questions are addressed in this book from a deterministic
approach. Quantitative answers are given through the development of
physical models that describe the optical process of image formation on the
fovea, and the subsequent neural processing of visual information gathered
by photoreceptors.

A faithful and robust simulation of the optical and visual performance
of the human eye is provided for axial vision of distant objects in a
variety of visual conditions. The book moves from intrinsically theoretical
aspects (optical and neurophysical models of the eye) to include a large
number of experimental measurements from within scientific literature.
The model parameters are tuned to the observed phenomenology, in order
to validate the predictive power of the models. The results turn out to be
very satisfactory in terms of quantitative and qualitative adherence of the
model predictions to field measurements.

The majority of material in this book is original and is the result of
investigations made by the author during the last decade. The most relevant
achievement of this work is the capacity to evaluate visual acuity for a
range of visual conditions, such as variations in pupil size, refractive error,
and ambient illumination.

The material is organized into two parts: optical and neurophysical
aspects of the eye model. Each part is then divided into two sections.
The first sections are devoted to assessment of the specific models
through derivation of parameters from the best-fitting of experimental
data. The second sections contain descriptions of the relevant properties
derived from the models, together with discussions and connections to
real-life situations. The reader should note that chapters and paragraphs
with high-level mathematical and physical optics content can be safely
skipped without compromising overall comprehension. To this end, a brief
summary is provided at the end of each chapter.

Part IA defines the optical eye model that is used throughout the book—
the chromatic aspherical Gullstrand exact (CAGE) eye model, which is
developed from the Gullstrand exact eye model with the introduction of
aspherical interfaces and chromatic index dispersion. Surface asphericities
are derived from the best-fitting of line images recorded in a classical
double-pass experiment, with similar images obtained from the CAGE
model. Theoretical modeling of the double-pass experiment requires
a complex physical optics analysis, including directionality of foveal
reflection and spatial partial coherence of illumination light. The
procedure is supported by the available accurate reporting of experimental
conditions. The result is an excellent match-up of model predictions with
measurements at all pupil sizes (R2 > 0.92). The values of surface
asphericities match well with independent measurements performed in
vivo.

Part IA demonstrates the feasibility of using schematic eye models
not only for estimating first-order geometrical optics properties and
aberrations, but also for evaluating and reproducing the actual retinal
images recorded by human eyes with high accuracy. The physical optics
approach is attractive, since the starting point for the calculation is not the usual wave aberration at the exit pupil (estimated from aberration data), but
a well-defined optical scheme. This approach allows for the joint treatment
of monochromatic and chromatic aberrations, as well as diffraction. As a
consequence, the CAGE model is representative of the average human eye
for distance foveal imaging.

Part IB provides a detailed presentation of optical performances
exhibited by the CAGE model. The model’s paraxial properties at the
central wavelength coincide with those of the Gullstrand exact model,
but vary with wavelength. The CAGE eye model is characterized through
the analysis of spherical aberration, point and line spread functions at
variable pupil sizes, relative energy content, and modulation transfer
function. Single-valued parameters are extracted for a simpler, direct
description of optical behavior, including Strehl and Struve ratios,
optimum defocus, full widths at half maximum for point and line images,
spatial frequency bandwidths, and retinal gain. The entire characterization
is illustrated by the continuous comparison between monochromatic and
white light performances, as well as by comparison with two diverging
behaviors: the diffraction-limited model and the purely spherical model
(Gullstrand exact). CAGE model predictions are successfully compared
with independent in-vivo measurements of spherical aberration and
psychophysical modulation transfer function.

The most important innovative contributions from Part IB are as
follows:
Optimum defocus is effective in maximizing the foveal performance against spherical aberration (explaining the hyperopic choice operated by Gullstrand in his model).
Retinal gain in conditions of optimum defocus is much larger than that assumed in international standards for laser safety.
Chromatic aberration is the major limiting factor of optical performance.
The eye behaves as a poor optical system in monochromatic illumination, but in white light it performs only 50% worse than a diffraction-limited eye.

In Part IIA, the CAGE optical eye model is merged with a neurophysical
model of the eye from Barten, which describes the psychophysical
response of the eye to sinusoidal bar stimuli with variable frequency,
contrast, and luminance (ocular contrast sensitivity). The Barten model
is based on the estimate of noise level generated internally in the eye. It
depends on a few scalar parameters related to the integration properties of
the eye, and on the ocular modulation transfer function. Modifications to
the original Barten model have been introduced for physical consistency
and improved phenomenological representation. The main modification involves the modulation transfer function of the eye, which is calculated
by means of the CAGE optical model. The joint CAGE-Barten model can
provide estimates of the contrast sensitivity function (CSF) for a wide
range of ambient and subject conditions. Values of the model parameters
are derived from the best-fitting of 15 experimental data series on CSF,
taken from the literature. The overall agreement obtained is excellent
(R2 > 0:96), providing good predictability in a variety of test conditions.

The main achievement of Part IIA is the development of a physical
model that can predict human contrast sensitivity for a large number
of conditions (including pupil size and refractive error of the subject;
spatial frequency, spectrum, size, and duration of the stimulus; and
ambient luminance). Results are obtained by following a deterministic
physical pathway, without any ad-hoc heuristic assumptions (as in
the original Barten model). Furthermore, values of the psychophysical
parameters (obtained from the best-fitting procedure) help to define both
structural properties of the eye (photoreceptor quantum efficiency, neural
noise spectral density) and features of the integration capability of the
visual system (temporal, spatial, and frequency integration limits, lateral
inhibition cutoff). Thus, the CAGE-Barten model represents an effective
tool for evaluating optical and perceptive properties of the human visual
system.

In Part IIB, visual performances of the CAGE-Barten model are
analyzed, starting from the evaluation of the entire perceptive region in
the contrast-spatial frequency plane, which characterizes the quality of
vision for any visual condition. The analysis is based on two single-valued
parameters—grating visual acuity and bilogarithmic area of the perceptive
region—which are evaluated as a function of pupil size and pupil response,
illumination spectrum, spherical aberration, defocus, stimulus properties,
and psychophysical parameters. The results are satisfactorily compared
with the experimental measures of Snellen visual acuity and image quality.
As an example, model grating visual acuity at 3.3-mm pupil size and 160-cd=m2 luminance is -0.14 logMAR (20/14.5 Snellen fraction), which well
overlaps with analogous measurements performed in young subjects. The
CAGE-Barten model allows analysis of visual performance in relation to
the fundamental limits placed by diffraction and noise, thus quantifying
potential margins of improvement. Despite being based on a single filterdetector
unit, the CAGE-Barten model is compatible with the existence of
a plurality of spatial frequency channels; also, fitting such channels into
the CSF evaluated by the model helps to shed light on their nature and
structure.

The main contribution of Part IIB is unification of the optical and
psychophysical descriptions of vision under a single model, with high
predictability of mean performances in the human eye. In addition to providing access to the neural image, the model provides local
and integrated metrics for the quantitative evaluation of vision quality,
related to variations of observing conditions. The CAGE–Barten model
represents an effective tool for reproducing and analyzing both imaging
and perception behaviors of an average human eye.

I am indebted to Dr. Laura Galli, Scientific Institute Hospital San
Raffaele, for precious statistical advice. I thank Prof. Gianni Gilardi,
Department of Mathematics F. Casorati, University of Pavia, for providing
me with useful analytical formulas. Finally, I am grateful to my wife Mara
and my children Alessandra and Francesco for their confident and patient
waiting for this laborious delivery.